We will accept the Rules of Reasoning that we have been discussing as our logic rules in our mathematical system. There are several ways in which we can go about setting forth our axioms and undefined terms for geometry. We can keep the number of axioms down to a very few, in which case we need a large number of undefined terms, relations, and operations. We can keep the number of undefined terms, relations, and operations to a small number, thus requiring a larger number of axioms. We will actually work with the Axiom system set up by David Hilbert in 1899 in his book Foundations of Geometry. He not only clarified Euclid's definitions but also filled in the gaps in some of Euclid's proofs. Hilbert recognized that Euclid's proof for the side-angle-side criterion of congruence in triangles was based on an unstated assumption (the principle of superposition) and that this criterion had to be treated as an axiom. He built on the earlier work of Moritz Pasch, who in 1882 published the first rigorous treatise on geometry; Pasch made explicit Euclid's unstated assumptions about betweenness. At the same time we could have used the axioms set forth by Garret Birkhoff. These axioms are the ones with which you are more familiar, being the basis for the texts in High School Geometry. I have included both sets of axioms, so that you can compare the two. We will work exclusively with the Hilbert axioms. An interesting exercise is to prove everything that we will prove using Birkhoff's axioms. Some proofs are easier, and some are harder.
First, let us set our undefined terms. They will be
These objects are to be taken as undefined terms. We will not attempt to make a definition of any of them. However, when we wish to look at a specific example of our geometry, called a model, we will have to establish how each of these undefined terms is to be interpreted. For example, if we wish to look at the model where points are ordered pairs coming from the usual Cartesian plane and lines are the usual straight lines in the Cartesian plane, then we would be able to easily state what it means for a point to be incident with a particular line, what it means for one point to lie between two other points and what it means for two line segments to be congruent. We could change our example though to be a set of seven points, where lines would be ordered pairs of points, incidence would be set inclusion and betweenness and congruence would be meaningless. The particular model that we choose will determine our interpretation of the terms, but will not define them.
At this point we need to make some definitions. In order to do this we need our axiom system. I shall include all of the axioms here, though we will discuss them in more detail later.
Hilbert's Axioms for Neutral Geometry
that passes through P and Q.
there exist at least two distinct
points incident with .
, then A, B, and C are three distinct
points all lying on the same line, and 
.
such that
, 
, and 
.
and
for any three points A, B, and C not lying on :
and B and
C are on the same side of , then A and C are on the same side of
.
and B
and C are on opposite sides of , then A and C are on the same
side of .
and 
.
and 
, then 
.
Moreover, every segment is congruent to itself.
, A'*B'*C',
, and 
, then 
.
and given any ray 
emanating
from a point A', then there is a unique ray 
on a given
side of line 
such that 
.
and 
,
then 
. Moreover, every angle is congruent to itself.
emanating from A, then a point E is reached where
.
is the union 
of two nonempty subsets
such that no point of 
is between two points of 
and
vice versa. Then there is a unique point, O, lying on such that
if and only if 
.
has one
point inside and one point outside another circle 
, then the two
circles intersect in two points.
Birkhoff's Axioms for Neutral Geometry
.
called
lines, with the property that each two points belong to exactly
one line.
there
exists a unique number 
, the
distance from A to B, which is 0 if and only if A = B.
) between the points 
and the
numbers 
such that
.
satisfying
. That is, they are pairwise disjoint.
then 
.
there exists a unique real
number x with 
which is the (degree) measure of the
angle 
, then
is a ray in the edge, k, of
an open half plane H(k;P) then there exist a one-to-one correspondence
between the open rays in H(k;P) emanating from A and the set of real
numbers between 0 and 180 so that if 
then
Definition: The plane is the collection of all points and lines.
Definition: The segment 
A and B are the endpoints of the
segment AB.
Definition: Given points O and A in the plane. The set of points P such
that 
is a circle with center O. Each segment OP
is called a radius .
Definition: The ray 
emanates from A and is
part of 
.
Definition: The rays and 
are opposite if they are
distinct, emanate from A, and are part of the same line
.
Definition: An angle with vertex A is
a point A together with
2 non-opposite rays and , called the sides ,
emanating from A. Denote this angle by 
.
Definition: If 
and 
have a common side 
and the other two sides and form opposite rays, the
angles are supplements or supplementary angles .
Definition: An angle is a right angle
if it has a supplementary angle to which it is congruent.
Definition: Two lines and m are parallel
if they do not intersect; i.e., if no point lies on both of them. Denote this by
.
Definition: Two lines and m are perpendicular
, 
,
if they have a point A in common and there exist rays , a part of
, and , a part of m, such that is a right
angle.